Snowflakes, Starflakes, and Swirlflakes Unusual variations on the paper snowflake.
Snowflakes, Starflakes, and Swirlflakes
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- So say it's the holiday season
- and you're supposed to be all festive and jolly.
- But you're more of the Grinchy type and
- all you want to do is wheel-trip objects against things.
- So you're doing your holiday part by making paper snowflakes.
- Regardless of your feelings about the holidays,
- slicing things into tiny bits is an art,
- and one that you are taking very seriously.
- Like, some people make so-called paper snowflakes
- by folding a piece of paper in half,
- and then in half again, and again,
- and then cutting it up.
- But paper snowflake connoisseurs know that
- real snowflakes have six-fold symmetry,
- and this thing has 4-fold symmetry;
- that would never happen in nature.
- So you do it the real way, by folding your paper in half twice,
- and then folding it in thirds.
- Or I suppose you could do it by halves, and then thirds,
- and then halves again.
- But you can't do thirds, then halves, then halves again,
- because what would that even mean?
- but you do notice your first halves can be at any angle you want -
- you don't need to line it up or anything.
- It's kind of funny, because to get six-fold symmetry,
- you need to fold it into 12 sections,
- and if you fold into 6 sections, you only get 3-fold symmetry.
- Which is actually a way that snowflakes occasionally form,
- so those are allowed.
- And then there are even sometimes 12-fold symmetric snowflakes in nature,
- which means you can fold again to make that - but never 4-fold.
- The problem with folding paper is
- that the thickness starts to get in the way.
- This makes points uneven, which might actually be more natural.
- Most real snowflakes are actually pretty lumpy and flawed -
- just, those aren't the ones people take and share pictures of.
- And that's not the kind of snowflake you want to make, either.
- I mean, when you fold this angle into thirds,
- this flap is under this one, so it has to be a little shorter -
- at least, if this edge lines up here.
- But maybe if you folded one in front and one in back,
- accordion style, then all three sections could be the same.
- In fact, then instead of folding it in half, like this,
- you could do each section back and forth, and that's much better.
- Or, maybe you get bored of 6-fold symmetry,
- and decide to make a 5-fold one.
- Well, if you need 5 lines of symmetry,
- that's 10 sections; so first you fold it in half,
- and then you need to fold this into 5ths.
- You can use a protractor, or just kind of eyeball it and adjust...
- There. 5-fold snowflake.
- In fact, if you get good at folding this initial 5-fold wedge,
- you can do a single straight cut on it to get a star super-quickly.
- Or you can slice it up and get lots of stars.
- Or cut fancy stuff in there for fancy snowflakes.
- Stars count as holiday spirit, right?
- And you can do 7-fold symmetry in a similar way,
- but you're probably gonna need your emergency protractor.
- But you can do 9-fold without a protractor,
- because you can do thirds and then thirds again,
- and if you can do 5ths without a protractor,
- you can do 10ths, too, because it's 1/5 times 1/2.
- Like I said, happy holidays; but I never said which ones.
- Valentines day is totally a winter holiday.
- 11 is prime, though, so time for the protractor again.
- Look, prime factorization!
- Okay, so now theoretically you can get all sorts of n-fold symmetry.
- But what about rotational symmetry?
- There's no mirror lines, which means no folding,
- so does it even make sense as a question?
- Cutting a snowflake design efficiently is
- all about putting the same cut lines on top of each other,
- so you only have to cut them once.
- So how do you take a rotationally symmetric design like this,
- and put all the layers on top of each other,
- without overlapping anything else?
- Maybe it's not surprising to see that
- to get stuff with rotational symmetry to line up, you rotate it.
- If you make a cut to the center, then you can rotate all the way,
- and roll the symmetry up into 1 unique thing.
- It's hard to draw accurate rotational symmetry by hand,
- but now I can symmetrize this badly-drawn Cyrell design.
- So to cut out a paper snowflake, start with a cut,
- then curl your paper into a cone.
- You can swirl around once, or twice, or more,
- but the important thing is to make sure the cut lines up with itself.
- Because as far as symmetry is concerned, that cut doesn't exist.
- I like to tape it in place so it doesn't unroll, then cut stuff out.
- I find that spirally things work well.
- Folding the paper is a good way to start a cut,
- but remember that folding creates symmetry,
- so I like to use it to just to get the scissors in there
- and then do something asymmetric. Voila! Swirlflake!
- For a starflake-swirlflake,
- you'll have to curl your paper around 5 times or 4 times?
- It's funny because I think of this as going around once,
- but really it's going around twice,
- and a flat sheet of paper goes around once.
- Anyway, yeah, do that, and then give it a nice spirally arm. Or two.
- You can make a nice fancy starflake-swirlflake-snowflake; awesome flake.
- Of course, from snowflakes
- it's only one small step to folding and cutting frieze patterns,
- and then wallpaper patterns,
- and hey, what kinds of patterns do you get?
- if you start by folding stuff into a mobius strip?
- And then maybe you'll want to start folding and cutting spheres,
- and everything will be a mess.
- So you'd better just stop. Now.
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